Lorenz gauge

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Looking at Maxwell's equations in terms of electromagnetic potentials,

\nabla^2 \Phi + \frac{\partial}{\partial t} (\boldsymbol{\nabla}\cdot\mathbf{A}) = -\frac{\rho}{\epsilon_0}\,,

and

\boldsymbol{\nabla} \left( \mu_0 \epsilon_0 \frac{\partial \Phi}{\partial t} + \boldsymbol{\nabla} \cdot \mathbf{A}\right) + \left( \mu_0 \epsilon_0\frac{\partial^2\mathbf{A}}{\partial t^2} - \nabla^2 \mathbf{A} \right) = \mu_0 \mathbf{J}\,,

we may perform a gauge transformation such that

\mu_0 \epsilon_0 \frac{\partial \Phi}{\partial t} + \boldsymbol{\nabla} \cdot \mathbf{A} = 0\,.

This is known as the Lorenz gauge, named after Danish physicist Ludvig Lorenz. This leads to the following wave equations:

\frac{1}{c^2}\frac{\partial^2 \Phi }{\partial t^2} - \nabla^2 \Phi = \frac{\rho}{\epsilon_0}\,,

\frac{1}{c^2} \frac{\partial^2\mathbf{A}}{\partial t^2} - \nabla^2 \mathbf{A} = \mu_0 \mathbf{J}\,.

Covariant form

\partial_\mu A^\mu = 0\,
back to gauge transformation
on to Coulomb gauge

See also


References

[1]

  1. ↑ Lorenz, L. (1867). "On the Identity of the Vibrations of Light with Electrical Currents". Philos. Mag. 34: 287-301. 
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